Timeline for Non-compact Dirichlet fundamental domains and free Fuchsian groups
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 24, 2021 at 15:12 | comment | added | Sam Nead | Yes, still negative. I've updated my answer. | |
| Aug 24, 2021 at 15:12 | history | edited | Sam Nead | CC BY-SA 4.0 |
answered (3) updated.
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| Aug 15, 2021 at 14:36 | comment | added | JackTodd | Thanks. I have accepted this is an answer. I have slightly weakened my statement in 3). Is the answer still negative? | |
| Aug 15, 2021 at 14:33 | vote | accept | JackTodd | ||
| Aug 15, 2021 at 8:59 | history | edited | Sam Nead | CC BY-SA 4.0 |
fixed answer to (1)
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| Aug 15, 2021 at 8:50 | comment | added | Sam Nead | Ah, I realise that I misread the intent of your question (1). I'll rewrite my answer. | |
| Aug 15, 2021 at 8:49 | comment | added | Sam Nead | There are no relations in that group. And yes, a fundamental domain is given by a pair of triangles. But that fundamental domain is not the Dirichlet domain based at the center of one of the triangles... | |
| Aug 15, 2021 at 6:48 | comment | added | JackTodd | It seems to me that your counterexample to my statement in (2) is given by the Fuchsian group associated with a $(\infty,\infty,\infty)$ triangle group. A fundamental domain is given by the union of two adjacent ideal triangles from the tessellation. But, I don't see that there are any relations in this Fuchsian group. So, (2) seems to hold true. If my understanding is wrong, it would be great if you could rephrase your counterexample in more elementary terms. | |
| Aug 15, 2021 at 2:10 | comment | added | Sam Nead | I assume you are asking about the double of an ideal triangle. Let $p_0$ and $p_1$ be the centres of the two triangles $T_0$ and $T_1$. Suppose that we place the basepoint of our Dirichlet domain at $p_0$. Let $L_j$ be the three geodesic rays in $T_1$ that emanate from $p_1$ and "end" at the three ideal points of $T_1$. We cut the surface along $\cup L_j$ to obtain the Dirichlet domain. It helps to draw a picture! | |
| Aug 15, 2021 at 1:36 | comment | added | JackTodd | ad 2) Could you explain the side-pairings of the Dirichlet fundamental domain in your example? Where are the 3 material vertices coming from? (PS: Thank you for your answer and comments.) | |
| Aug 14, 2021 at 16:28 | comment | added | Sam Nead | By the way, it is polite to upvote an answer if you find it helpful. And, if it answers your question, then you should "accept" it (by ticking the tick mark). | |
| Aug 14, 2021 at 16:24 | comment | added | Sam Nead | Let $T$ be an ideal triangle. Define $T_i = T \times \{i\}$ for $i = 0, 1$. So each of $T_0$ and $T_1$ are also ideal triangles. We define a new topological space $S$ by gluing $(x, 0) \in T_0$ to $(x, 1) \in T_1$ if and only if $x$ is a boundary point of $T$. It is now an exercise to define the natural hyperbolic structure on $S$. (In fact, there is a subtle point here about the induced orientation on $S$. But I think it is fair to simplify the exposition for the first presentation of the idea.) | |
| Aug 14, 2021 at 14:41 | comment | added | JackTodd | Could you please clarify "surface made by doubling an ideal triangle across its boundary"? | |
| Aug 14, 2021 at 13:28 | history | edited | Sam Nead | CC BY-SA 4.0 |
Added further suggestion about how to get a "positive" result.
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| Aug 14, 2021 at 12:39 | history | answered | Sam Nead | CC BY-SA 4.0 |